Subgroups — #31912372

Order 18176.

586 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556068	C₂	1	1	yes	maximal, characteristic
2	C₂×#17986037	C₂	1	1	yes	maximal, characteristic
3	#22586447	C₂	1	4	yes	maximal, complemented
4	C₂×#427×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556087	—	2	1	no	—
6	C₂×#14556068	—	2	1	no	—
7	C₂×#14556068	—	2	2	no	—
8	C₂×#14556068	C₄	1	4	yes	complemented
9	C₂×#14556068	C₂²	1	2	yes	—
10	C₂²×C₇₁⋊C₁₆	—	2	1	no	—
11	#17986037	—	2	1	no	—
12	#17986037	—	2	1	no	—
13	#17986037	C₄	1	2	yes	complemented
14	#17986037	C₂²	1	2	yes	—
15	C₂×#14556062	C₄	1	2	yes	complemented
16	C₂×#14556060	—	2	1	no	—
17	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
18	C₂×#117×C₇₁	—	2	1	no	—
19	#427×C₇₁	—	2	2	no	—
20	#427×C₇₁	C₄	1	4	yes	complemented
21	C₂×#107×C₇₁	—	2	1	no	—
22	C₂×#106×C₇₁	C₂²	1	2	yes	—
23	#14556087	—	2	2	no	—
24	#14556087	—	2	2	no	—
25	#14556087	—	4	1	no	—
26	#14556087	—	4	1	no	—
27	C₂×C₄⋊Dic₇₁	—	2	2	no	—
28	C₂×C₂²⋊Dic₇₁	—	4	1	no	—
29	C₂×C₄×Dic₇₁	—	4	1	no	—
30	#14556068	—	2	4	no	—
31	#14556068	—	2	2	no	—
32	#14556068	C₂×C₄	1	4	yes	complemented
33	C₂×C₇₁⋊C₁₆	—	2	4	no	—
34	C₂×C₇₁⋊C₁₆	—	2	2	no	—
35	#14556062	—	2	2	no	—
36	#14556062	C₂×C₄	1	4	yes	—
37	#14556060	—	2	2	no	—
38	#14556060	—	2	2	no	—
39	C₂³×C₄×C₇₁	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₇₁	D₄	1	2	yes	—
48	C₂²×C₈×C₇₁	Q₈	1	2	yes	—
49	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
50	#117×C₇₁	—	2	4	no	—
51	#117×C₇₁	—	4	2	no	—
52	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
53	C₂×C₂²⋊C₄×C₇₁	—	4	1	no	—
54	C₂×C₄²×C₇₁	—	4	1	no	—
55	#107×C₇₁	—	2	4	no	—
56	#106×C₇₁	—	2	2	no	—
57	#106×C₇₁	C₂×C₄	1	4	yes	complemented
58	C₂²×Dic₇₁	—	4	2	no	—
59	C₄⋊Dic₇₁	—	2	4	no	—
60	C₄⋊Dic₇₁	—	4	2	no	—
61	C₂²⋊Dic₇₁	—	4	2	no	—
62	C₂²⋊Dic₇₁	—	4	2	no	—
63	C₄×Dic₇₁	—	4	2	no	—
64	C₄×Dic₇₁	—	4	2	no	—
65	C₇₁⋊C₁₆	—	2	4	no	—
66	C₂⁴×C₇₁	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₇₁	—	4	2	no	—
68	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
74	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
75	C₂²×C₄×C₇₁	SD₁₆	1	2	yes	—
76	C₂²×C₄×C₇₁	Q₁₆	1	2	yes	—
77	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₇₁	C₄²	1	2	yes	—
79	C₂×C₈×C₇₁	C₄²	1	2	yes	—
80	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
83	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
84	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
85	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
86	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
87	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
88	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
89	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
90	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
96	C₄⋊C₄×C₇₁	—	2	4	no	—
97	C₄⋊C₄×C₇₁	—	4	2	no	—
98	C₂²⋊C₄×C₇₁	—	4	4	no	—
99	C₄²×C₇₁	—	4	4	no	—
100	C₂³×C₇₁	—	4	2	no	—
101	C₂³×C₇₁	#102	1	1	yes	characteristic
102	C₂³×C₇₁	#102	1	1	yes	characteristic
103	C₂³×C₇₁	#102	1	2	yes	—
104	C₂³×C₇₁	#107	1	2	yes	—
105	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₇₁	—	4	4	no	—
107	C₂×C₄×C₇₁	—	8	4	no	—
108	C₂×C₄×C₇₁	#95	1	2	yes	—
109	C₂×C₄×C₇₁	#95	1	2	yes	—
110	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
111	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
112	C₂×C₄×C₇₁	#102	1	2	yes	—
113	C₂×C₄×C₇₁	#103	1	2	yes	—
114	C₂×C₄×C₇₁	#103	1	2	yes	—
115	C₂×C₄×C₇₁	#103	1	2	yes	—
116	C₂×C₄×C₇₁	#103	1	2	yes	—
117	C₂×C₄×C₇₁	#106	1	2	yes	—
118	C₂×C₄×C₇₁	#106	1	2	yes	—
119	C₂×C₄×C₇₁	#107	1	2	yes	—
120	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
126	C₂×Dic₇₁	—	4	4	no	—
127	C₂×Dic₇₁	—	8	4	no	—
128	C₈×C₇₁	#95	1	2	yes	—
129	C₈×C₇₁	#95	1	2	yes	—
130	C₈×C₇₁	C₂×C₄²	1	1	yes	characteristic
131	C₈×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂²×C₇₁	—	2	2	no	—
135	C₂²×C₇₁	—	2	2	no	—
136	C₂²×C₇₁	—	2	2	no	—
137	C₂²×C₇₁	—	4	2	no	—
138	C₂²×C₇₁	—	4	2	no	—
139	C₂²×C₇₁	#340	1	2	yes	—
140	C₂²×C₇₁	#340	1	2	yes	—
141	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
142	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
143	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
144	C₄×C₇₁	—	8	4	no	—
145	C₄×C₇₁	#340	1	2	yes	—
146	C₄×C₇₁	#340	1	2	yes	—
147	C₄×C₇₁	C₂×#95	1	1	yes	characteristic
148	C₄×C₇₁	C₂×#103	1	1	yes	characteristic
149	C₄×C₇₁	C₂×#103	1	1	yes	characteristic
150	C₄×C₇₁	C₂×#106	1	1	yes	characteristic
151	Dic₇₁	—	8	4	no	—
152	#12567	—	71	1	no	maximal, Sylow
153	C₂×C₇₁	—	2	1	no	—
154	C₂×C₇₁	—	2	2	no	—
155	C₂×C₇₁	—	2	2	no	—
156	C₂×C₇₁	—	2	1	no	—
157	C₂×C₇₁	C₂×#340	1	1	yes	characteristic
158	C₂×C₇₁	#2145	1	2	yes	—
159	C₂²×M₃₂	—	71	1	no	—
160	C₂×#427	—	71	1	no	—
161	C₂×#427	D₇₁	1	1	yes	characteristic
162	#1386	—	71	4	no	—
163	C₇₁	#12567	1	1	yes	characteristic, complemented, Sylow
164	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
165	C₂×#117	—	2	1	no	—
166	C₂×#117	—	142	1	no	—
167	C₂×M₃₂	—	71	4	no	—
168	C₂×M₃₂	—	71	2	no	—
169	C₂×M₃₂	—	142	1	no	—
170	C₂×M₃₂	—	142	2	no	—
171	C₂²×C₁₆	—	142	1	no	—
172	#427	—	2	2	no	—
173	#427	—	71	2	no	—
174	#427	—	71	2	no	—
175	#427	—	142	1	no	—
176	#427	—	142	1	no	—
177	#427	Dic₇₁	1	4	yes	complemented
178	C₂×#107	—	2	1	no	—
179	C₂×#107	—	71	2	no	—
180	C₂×#106	—	142	1	no	—
181	C₂×#106	C₂×D₇₁	1	2	yes	—
182	C₂³×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
183	C₂²×C₈	Dic₁₄₂	1	2	yes	—
184	C₂²×C₈	C₄×D₇₁	1	2	yes	—
185	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
186	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
187	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
188	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
189	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
190	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
191	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
192	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
193	#117	—	2	4	no	—
194	#117	—	4	2	no	—
195	#117	—	284	1	no	—
196	#117	—	284	1	no	—
197	#117	—	142	2	no	—
198	#117	—	142	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	142	2	no	—
201	C₂×C₂²⋊C₄	—	4	1	no	—
202	C₂×C₂²⋊C₄	—	284	1	no	—
203	C₂×C₄²	—	4	1	no	—
204	C₂×C₄²	—	284	1	no	—
205	M₃₂	—	71	4	no	—
206	M₃₂	—	142	4	no	—
207	M₃₂	—	142	2	no	—
208	C₂×C₁₆	—	142	4	no	—
209	C₂×C₁₆	—	142	2	no	—
210	#107	—	2	4	no	—
211	#107	—	71	4	no	—
212	#107	—	142	2	no	—
213	#106	—	2	2	no	—
214	#106	—	142	2	no	—
215	#106	—	142	2	no	—
216	#106	C₂×Dic₇₁	1	4	yes	complemented
217	C₂⁴	#11765850	1	1	yes	characteristic
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	—	284	2	no	—
220	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
221	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
222	C₂²×C₄	C₂²⋊Dic₇₁	1	2	yes	—
223	C₂²×C₄	Dic₇₁⋊C₄	1	2	yes	—
224	C₂²×C₄	#11765850	1	1	yes	characteristic
225	C₂²×C₄	#11765852	1	1	yes	characteristic
226	C₂²×C₄	#11765852	1	1	yes	characteristic
227	C₂²×C₄	#11765853	1	2	yes	—
228	C₂²×C₄	#11765855	1	2	yes	—
229	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
230	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
231	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
232	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
233	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
234	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
235	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
236	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
237	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
238	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
239	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
240	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
241	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
242	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
243	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
244	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
245	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
246	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
247	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	284	2	no	—
251	C₄⋊C₄	—	142	4	no	—
252	C₂²⋊C₄	—	4	4	no	—
253	C₂²⋊C₄	—	284	2	no	—
254	C₂²⋊C₄	—	284	2	no	—
255	C₄²	—	4	4	no	—
256	C₄²	—	284	2	no	—
257	C₄²	—	284	2	no	—
258	C₁₆	—	142	4	no	—
259	C₂³	—	4	2	no	—
260	C₂³	#14556048	1	1	yes	characteristic
261	C₂³	#14556048	1	1	yes	characteristic
262	C₂³	#14556049	1	2	yes	—
263	C₂³	#14556061	1	2	yes	—
264	C₂³	C₂×#11765850	1	1	yes	characteristic
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	8	4	no	—
267	C₂×C₄	—	284	4	no	—
268	C₂×C₄	—	568	4	no	—
269	C₂×C₄	C₇₁⋊#95	1	2	yes	—
270	C₂×C₄	C₇₁⋊#95	1	2	yes	—
271	C₂×C₄	#14556048	1	1	yes	characteristic
272	C₂×C₄	#14556048	1	1	yes	characteristic
273	C₂×C₄	#14556049	1	2	yes	—
274	C₂×C₄	#14556051	1	2	yes	—
275	C₂×C₄	#14556051	1	2	yes	—
276	C₂×C₄	#14556052	1	2	yes	—
277	C₂×C₄	#14556052	1	2	yes	—
278	C₂×C₄	#14556059	1	2	yes	—
279	C₂×C₄	#14556059	1	2	yes	—
280	C₂×C₄	#14556061	1	2	yes	—
281	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
282	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
283	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
284	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
285	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
286	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
287	C₈	C₇₁⋊#95	1	2	yes	—
288	C₈	C₇₁⋊#95	1	2	yes	—
289	C₈	C₂×C₄×Dic₇₁	1	1	yes	characteristic
290	C₈	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
291	C₈	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
292	C₈	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	4	2	no	—
297	C₂²	—	4	2	no	—
298	C₂²	#17985761	1	2	yes	—
299	C₂²	#17985761	1	2	yes	—
300	C₂²	C₂×#14556048	1	1	yes	characteristic
301	C₂²	C₂×#14556049	1	1	yes	characteristic
302	C₂²	C₂×#14556061	1	1	yes	characteristic
303	C₄	—	8	4	no	—
304	C₄	—	568	4	no	—
305	C₄	#17985761	1	2	yes	—
306	C₄	#17985761	1	2	yes	—
307	C₄	C₂×C₇₁⋊#95	1	1	yes	characteristic
308	C₄	C₂×#14556051	1	1	yes	characteristic
309	C₄	C₂×#14556052	1	1	yes	characteristic
310	C₄	C₂×#14556059	1	1	yes	characteristic
311	C₂	—	2	1	no	—
312	C₂	—	2	2	no	—
313	C₂	—	2	2	no	—
314	C₂	—	2	1	no	—
315	C₂	C₂×#17985761	1	1	yes	characteristic
316	C₂	#22590075	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14556048
commutator	C₈×C₇₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#427×C₇₁	C₂
socle	C₂²×C₇₁	C₂×#102
prime radical	C₂⁴×C₇₁	D₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#427	D₇₁
hypocenter	C₇₁	#12567
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #22586447	C₂	semidirect
8. C₂×#14556068	C₄	semidirect
13. #17986037	C₄	semidirect
15. C₂×#14556062	C₄	semidirect
20. #427×C₇₁	C₄	semidirect
32. #14556068	C₂×C₄	semidirect
57. #106×C₇₁	C₂×C₄	semidirect
163. C₇₁	#12567	semidirect
177. #427	Dic₇₁	semidirect
216. #106	C₂×Dic₇₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#14556068	2
C₂×#17986037	2
#22586447	2
C₂×#427×C₇₁	2
#12567	71